Playing with radioactive decay equations

NND 0

multiply each side by et substitute N=N0-D

teNN 0

teNDN 00

teNND 00

)1(0teND

0NNe t

DNNe t

substitute N0=N-D

NNeD t

)1( teND

U-series Disequilibrium 9/13/10

Lecture outline:

1) Secular equillibrium and disequilibrium

2) U-Th systematics

3) U-excess

4) U-Th disequilbrium dating

5) 232Th “initial” corrections

6) Th excess and sedimentation rates

Geological archives dated by U-series disequilibrium:speleothems (top) and fossil corals (bottom)

Zircon

Secular equilibrium and disequilibrium

Secular equilibrium: all radioactive species in a decay chain have the same activity

Disequilibrium: system is perturbed (removal/enrichment of daughter/parent), and system decays back to secular equilibriumAnd if you know how (disequilibrium)initial, can calculate t since disequilibrium

For U-series decay chain, what are some examples of

processes that cause disequilibrium ?

There are two types of disequilibria:1) Daughter excess (i.e. activity of daughter > activity of parent)2) Daughter deficit (Ad < Ap)

N11 N2

2 N3

dN1dt

1N1dN2dt

1N1 2N2dN3dt

2N2

Decay Chain Systematics:

Consider a 3-member decay chain:

Evolution of this system is governed by the coupled equations:

Note that at secular equil, 2d0

d

N

t

As you can see, the solution of these differential equations is quite complicated (except for N1), so we will derive some equations from the disequilibria of 234U and 230Th.

N2 (t)1

2 1N1o e 1t e 2t N2oe 2t

234U Excess

Given: (234U/238U)A of ocean = 1.15Explanation: [you tell me]

The activity of (234U)excess decreases with time: 234234 234 0 tEx ExU U e

And excess 234U corresponds to the 234U not supported by 238U:

234234 238 234 0 238( ) tU U U U e

NOTE: everything in thislecture will be activities (A),unless otherwise noted

And dividing through by 238U activity, we obtain:234

234 0 238234

238 2381 t

A

U UUe

U U

So if you measure 234U/238U,and know (234U/238U)initial can calculate age

Note that fixed analytical error (±0.5%)yields larger and larger age error barsas you approach secular equilibrium

230Th Deficiency

Given: Many U-rich minerals (such as carbonate) precipitated with virtually no ThExplanation: [you tell me]

So you grow in 230Th due to decay of 238U and excess 234U (in atom number):

1 2 212 1 2

2 1

( ) t t to oN t N e e N e

234 230230 234234

230 234

t toEx ExTh U e e

And converting to activity, substituting formula for 234UEx, dividing by 238UA,and simplifying, we obtain:

230 234 230

234 0230230

238 238230 234

(1 ) 1 ( )t t t

A A

UThe e e

U U

So if you measure (230Th/238U), and assume initial (234U/238U)A=1.15, then you can calculate a sample’s age.

*Or, more realistically, you measure (230Th/238U) and 234U/238U,and iteratively find an age that satisfies both the measurementsmade today. You then are calculating also 234U/238U initial.

So when/where is the assumption that

initial (234U/238U)A=1.15not a good one?

secularequilibrium

secularequilibrium

230Th-234U activity growth lines

For most samples:

Development of mass spectrometry techniques enable U-Th ages to be measured to ±0.1-5 precisions (Edwards et al., 1987)

Common U-Th series applications:

1. Corals- sea level from fossil terraces- climate reconstruction

2. Cave Stalagmites- climate reconstruction

Cutler et al., 2003

Edwards et al., 1987

but what happens when good samples get “dirty”?

FACT: most geological samples contain some “initial” or “detrital” or “nonradiogenic” thorium

PROBLEM: mass specs cannot distinguish between “detrital” 230Th and radiogenic 230Th

SAVING GRACE: “detrital” thorium mostly 232Th (quasi-”stable” on U/Th disequilibrium timescale)

STRATEGY: measure 232Th in samples, correct for detrital 230Th using 230Th/232Thof contaminant

BUT how do you estimate 230Th/232Th of contaminant?

given: average bulk Earth abundance (230Th/232Th)atom = 4.4e-6at secular equilibrium (Kaufman, 1993)

complication: in most settings this will not apply…

STRATEGY #1: date samples of known age

- especially good for coralsbecause you can absolutelydate them by counting backannual density bands(Cobb et al., 2003a)

-if you can identify a well-datedevent in your sample (volcaniceruption from historic record?), you may be able to do this for older samples

lower error bar = analytical error

upper error bar =analytical error + correctionusing (230Th/232Th)atom of 2.0e-5

STRATEGY #2: generate isochrons from samples

IDEA: sample multiple samples of the same age but different 232Th concentrations,then you know that they all contain the same 230Thrad, and that 230Thnr

will scale with 232Th

“dirty” edges

“clean” middle

“dirty” edges

U/Th isochron plots

- most often used forstalagmites (Partin et al., 2007)

- in these plots, slope α age intercept = (230Th/232Th)act

of contaminant phase

Usually huge range of values uncovered…

translating into huge age errors for “dirty” samplesLESSON: keep it clean (if possible)

Phenomenon: “excess” 230Th present in ocean sedimentsExplanation?

The activity of (230Th)excess decreases with time:

230Th Excess and Deep-Sea sediments

230230 230 0 tEx ExTh Th e

We can define sedimentation rate as:S=distance/time; so t=d/Sand

230 ( / )230 230 0 d SEx ExTh Th e

230 230 0230ln( ) ln( ) ( )Ex Ex

dTh Th

S

and

Depth in sediment core

ln(23

0 Th)

ex

Slope = -/S

If you assume that delivery of 230ThEx is constant through time, can calculateage as a function of depth - or - sedimentation rate

230Th Excess and sedimentation rate changes

230Th Excess - interpretation

Down-core measurementsin Norwegian core showsedimentation rate changes

But how can you have an increase of 230Thex with depth, given constant 230Thex delivery?

So 230Thex delivery is notconstant.Changes in the rain-rate of particles will lead to increasesand decreases in Th scavenging.

Increased scavengingof 230Th during interglacials in Norway,or sediment focusing.

What could change scavenging rate?